\documentclass[12pt]{amsart}
\usepackage{amssymb,amscd,amsmath,amsthm,color}
\usepackage{calc}
\usepackage{amsrefs}
\usepackage{color}
\usepackage[T1]{fontenc}
\usepackage{mathtools}
\usepackage[normalem]{ulem}
\usepackage{tikz}
\usetikzlibrary{matrix,arrows,decorations.pathmorphing}
\usepackage{setspace}
\usepackage{verbatim}
\usepackage{mathrsfs}
%\usepackage[notcite, notref]{showkeys}
%\usepackage[left=2cm,top=2cm,right=2cm,nohead]{geometry}

\textwidth 6.5 in
\oddsidemargin 0 in
\evensidemargin 0 in
\topmargin -.125 in
\textheight 8.75 in


\definecolor{bettergreen}{rgb}{0,.7,0}
%\newcommand\blue[1]{{\color{blue}{#1}}}
%\newcommand\green[1]{{\color{bettergreen}{#1}}}
%\newcommand\red[1]{{\color{red}{#1}}}

%\long\def\comment#1{\marginpar{{\footnotesize\color{red} #1\par}}}
%\long\def\commentimmi#1{\marginpar{{\footnotesize\color{bettergreen} #1\par}}}
%\long\def\change#1{{\color{blue} #1}}
%newcommand\green[1]{{\color{green} #1}}
%\newcommand   [1]{{\color{red}\small #1}}

%\let\immi=\green
%\let\raf=\blue

\newcommand\todo[1]{\ \vspace{5mm}\par \noindent\framebox{\begin{minipage}[c]{0.95 \textwidth} \tt #1\end{minipage}} \vspace{5mm} \par}


\DeclarePairedDelimiter\ceil{\lceil}{\rceil}
\DeclarePairedDelimiter\floor{\lfloor}{\rfloor}


%%%%  Definitions  re: typeface  %%%%%%%%%%

\newcommand{\A}{\mathbb{A}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\N}{{\mathbb N}}
\newcommand{\LL}{{\mathbb L}}
\newcommand{\TT}{{\mathbb T}}
\newcommand{\CC}{{\mathbb C}}
\newcommand{\ZZ}{{\mathbb Z}}


\newcommand{\bF}{\mathbf{F}}
\newcommand{\Gl}{\mathbf {GL}}
\newcommand{\SL}{\mathbf {SL}}
\newcommand{\Sp}{\mathbf {Sp}}
\newcommand{\aut}{\mathbf{Aut}}
\newcommand{\bG}{\mathbf{G}}
\newcommand{\bT}{\mathbf {T}}
\newcommand{\bM}{\mathbf {M}}


\newcommand{\cF}{\mathcal{F}}
\newcommand{\cO}{{\mathcal O}}
\newcommand{\ri}{\mathcal{O}}
\newcommand{\cM}{\mathcal {M}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\cG}{\mathcal{G}}
\newcommand{\cB}{\mathcal{B}}


\newcommand{\gl}{\mathfrak{gl}}
\newcommand{\fg}{\mathfrak{g}}
\newcommand{\ft}{\mathfrak{t}}
\newcommand{\fz}{\mathfrak{z}}
\newcommand{\fsp}{\mathfrak{sp}}
\newcommand{\fsl}{\mathfrak{sl}}
\newcommand{\fO}{\mathfrak{O}}
\newcommand{\fP}{\mathfrak{P}}
\newcommand{\ff}{\mathfrak{f}}
\newcommand{\fh}{\mathfrak{h}}
\newcommand{\fF}{\mathfrak{F}}


\newcommand{\reg}{\mathrm{rss}}
\newcommand{\ur}{\mathrm{ur}}
\newcommand{\nil}{\mathrm{Nil}}
\newcommand{\val}{\mathrm{val}}
\newcommand{\rs}{\mathrm{rs}}
\newcommand{\ram}{\mathrm{Ram}}
\newcommand{\can}{\mathrm{can}}
\newcommand{\diag}{\mathrm{diag}}
\newcommand{\oi}{{\bf \mathrm{O}}}


\newcommand{\Ad}{\operatorname{Ad}}
\newcommand{\jac}{\operatorname{Jac}}
\newcommand{\gal}{\operatorname{Gal}}
\newcommand{\res}{\operatorname{Res}}
\newcommand{\vol}{\operatorname{vol}}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\ep}{\operatorname{EP}}
\newcommand{\cI}{{\mathcal I}}


\newcommand{\rf}{k}


%%%%  Motivic  definitions  %%%%%%%

\newcommand{\ldp}{{\mathcal L}_{\mathrm {DP}}}
\newcommand\ldpo[1][\ri]{{\mathcal L}_{#1}}
\newcommand\cmf[1]{{\mathcal C}(#1)}
\newcommand\cA{{\mathcal A}}
%\newcommand\co{{\mathcal O}}
\newcommand\ord{\mathrm{ord}}
\newcommand\ac{\overline{\mathrm{ac}}}
\newcommand\lef{\mathbb L}
\newcommand\cP{{\mathcal P}}
\newcommand\cC{{\mathcal C}}
\newcommand\cH{{\mathcal H}}
\newcommand\cX{{\mathcal X}}
\newcommand\mot{\mathrm{mot}}
\newcommand{\scD}{\mathscr{D}}
\newcommand{\rss}{\mathrm{rss}}
\newcommand{\bGam}{{\mathbf \Gamma}}
\newcommand{\Loc}{\mathrm{Loc}}


\newcommand{\de}{{\text{Def}}}
\newcommand{\rde}{{\text{RDef}}}
\newcommand{\K}{F}
\newcommand{\mexp}{\mathbf{e}}
\newcommand{\tf}{{C_c^\infty}}


\def\llp{\mathopen{(\!(}}
\def\llb{\mathopen{[\![}}
\def\rrp{\mathopen{)\!)}}
\def\rrb{\mathopen{]\!]}}


%%%%%%%%Definitions from Lance %%%%%%%%%%%%%%%%

\newcommand{\Hasse}{\operatorname{Hasse}}
 %\newcommand{\diag}{\operatorname{diag}}
 \newcommand{\spa}{\operatorname{span}}
 \newcommand{\spec}{\operatorname{spec}}
 \newcommand{\Stab}{\operatorname{Stab}}
\newcommand{\m}{\operatorname{mod}}
 \newcommand{\tr}{\operatorname{tr}}
 \newcommand{\disc}{\operatorname{disc}}
 \newcommand{\meas}{\operatorname{meas}}
 \def\ve{\varepsilon}
 \def\an{_{\text{aniso}}}
 \def\gerg{\mathfrak{g}}
\def\gerh{\mathfrak{h}}
\def\gert{\mathfrak{t}}
\def\gergbar{\overline{\mathfrak{g}}}
\def\dgdot{d\dot{g}}
\def\dmu{d\mu}
\def\Qtilde{\widetilde{Q}}
%\def\cA{\mathcal{A}}
\def\C{\mathbb C}
\def\R{\mathbb R}
\def\N{\mathbb N}
\def\Q{\mathbb Q}
\def\Z{\mathbb Z}
\def\A{\mathbb A}
\def\Fp{\mathbb{F}_p}
\def\Fq{\mathbb{F}_q}
\def\Qp{\mathbb{Q}_p}
\def\GL{\mathrm{GL}}
\def\GLN{\mathrm{GL}_N}
\def\GLn{\mathrm{GL}_n}
\def\Lie{\mathrm{Lie}}
%\def\SL2{\mathrm{SL}_2}
%\def\Sp{\mathrm{Sp}}
\def\sln{\mathfrak{sl}_n}
\def\gln{\mathfrak{gl}_n}
\def\sl{\mathfrak{sl}}
\def\Gm{{\mathbb{G}_m}}
\def\Grs{G_{\text{rss}}}
\def\Gbar{\overline{G}}
%\def\bG{\mathbb{G}}
\def\bc{\boldsymbol{c}}
%\def\bT{\mathbb{T}}
\def\FpT{\mathbb{F}_p((T))}
\def\Fbar{\overline{F}}
\def\Aff{\mathbb{A}}
\def\cci{C_c^\infty}
\def\cO{\mathcal{O}}
\def\cQbar{\overline{\mathcal{Q}}}
\def\cQtilde{\widetilde{\mathcal{Q}}}
\def\spn{\mathfrak{sp}_{2n}}
\def\sp{\mathfrak{sp}}
\def\cT{\mathcal{T}}
\def\cL{\mathcal{L}}
\def\cP{\mathcal{P}}
\def\cN{\mathcal{N}}
\def\cE{\mathcal{E}}
\def\cF{\mathcal{F}}
\def\cH{\mathcal{H}}
\def\cR{\mathcal{R}}
\def\cS{\mathcal{S}}
\def\cQ{\mathcal{Q}}
\def\cX{\mathcal{X}}
\def\gO{\mathfrak{O}}
\def\gp{\mathfrak{p}}
\def\gP{\mathfrak{P}}
\def\gq{\mathfrak{q}}
\def\inv{^{-1}}

% Defs from Hales

\newcommand{\op}[1]{\operatorname{#1}}
\newcommand{\ring}[1]{{\mathbb #1}}
\def\rtie{\times}
\newcommand{\NF}{\op{NF}}

% More defs 

\newcommand{\fG}{\mathfrak G}
\newcommand{\fT}{\mathfrak T}

%%%%%%%%%%%%% Theorem declarations %%%%%%%%%%%%%

\theoremstyle{plain}
\newtheorem{thm}{Theorem}
\newtheorem{theorem}[thm]{Theorem}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
%\newtheorem{defn}[thm]{Definition}
%\newtheorem{rem}[thm]{Remark}
\newtheorem{prop}[thm]{Proposition}

\theoremstyle{definition}
\newtheorem{rem}[thm]{Remark}
\newtheorem{defn}[thm]{Definition}
\newtheorem{example}[thm]{Example}

\title{Coefficients of Harish-Chandra's local character expansion are motivic}

\author{Julia Gordon, Thomas Hales and Loren Spice}



\begin{document}

%\begin{abstract}  
%\end{abstract}

\maketitle

\rightline{-4919.5 hours}
\section{Introduction}
The present version of article is still inspired by \cite{unsuccessful self-treatment of writer's block}. 
 
We try to prove that the coefficients of Harish-Chandra's local character expansion are motivic functions of the parameters defining the group and the representation. 

\section{Reductive groups}
\subsection{Fixed choices and cocycle spaces}
Define field extensions, fixed data, the cocycle space $Z$, and a family of reductive groups $G$ over $Z$ as in \cite{transfer transfer}. Define Haar measure on $G_z$ the same way as well.
Will also use the term ``fixed choices'' in the same sense. 

\subsection{Nilpotent orbits} Let $\NF_G^k$ be the space of 
$k$-tuples of Barbasch-Moy pairs $\Upsilon=(N, \ff)$ as in  \cite{transfertransfer}. 
For all $F\in \Loc_m$ with sufficiently large $m$, 
a point in $\NF_G^k(F)$ gives a tuple of $(N_i, \ff_i)_{i=1}^k$ with $N_i$ representing distinct nilpotent orbits in $\fg(F)$, with $\ff_i$ being their corresponding facets in Barbasch-Moy classification. We also have a family of definable test functions $1_{\Upsilon}$ for 
$\Upsilon=(N, \ff)\in \NF_G^k$. 

%\subsection{orbits with a fixed semisimple part} 
We shall also need to consider the following situation. 
Let $G^0$ be a twisted Levi in $G$. 
Let $X_0$ be a central element in $\fg^0=\Lie(G^0)$. 
Then we will need a construction similar to the above for the set of $G^0$-orbits of elements whose semisimple part is $X_0$. These are in bijection with the nilpotent orbits in $G^0$ (see below). 
Let $\NF_{G^0}^k$ be the corresponding parametrizing set for these orbits. 

\section{Representations}
We assume that $p$ is large. 
Supercuspidal representations $\pi$ of $G_z$ are parametrized by J.-K. Yu's data 
$(\vec G, \vec\phi, \pi_0, x)$, where $\vec G=(G^0, \dots G^d)$ is a sequence of twisted Levi subgroups with $G^d=G$, $\phi_i$ is a character of $G^i$, $\pi_0$ is a depth-zero supercuspidal representation of $G^0$, and $x$ is a point in the reduced building of $G$.
Let us fix the depth of $\pi$. 

The groups $G^i$ are encoded by appropriate  cocycle spaces $Z^i$.

The representation $\pi_0$ is determined by its inducing datum: a parahoric subgroup $K^0$ and a representation $\rho_0$ of its reductive quotient $\fG^0$. 

\subsection{Parahorics}

\renewcommand\AA{\mathcal A}
\newcommand\affW{W_{\text{aff}}}
\newcommand\BB{\mathcal B}
	\newcommand\motB{\BB_{\text{mot}}}
	\newcommand\motBwith[1]{\BB_{\text{mot; #1}}}
\newcommand\bS{\mathbf S}
\newcommand\Cent{C}
	\newcommand\ldef{\mathrel{:=}}
\newcommand\dota{\cdot}
\newcommand\field{F}
	\newcommand\unfield{F^{\text{un}}}
\newcommand\Int{\operatorname{Int}}
\newcommand\Lxxx[1]{\ensuremath\spadesuit\footnote{\ensuremath\spadesuit\ #1}}
	\newcommand\citeme{\Lxxx{Cite me!}}
	\newcommand\refme{\Lxxx{Reference me!}}
\newcommand\muhat{\widehat\mu}
\newcommand\Nchar{\Phi}
\newcommand\NFTorbit{\widehat O}
\newcommand\Norm{N}
\DeclarePairedDelimiterX\pair[2]\langle\rangle{#1, #2}
\newcommand\Root{\Phi}
	\newcommand\AffRoot{\Psi}
\newcommand\Simple{\Delta}
\newcommand\st{\mathbin:}
\newcommand\stab{\operatorname{stab}}
	\newcommand\lsub{\prescript{}}
	\newcommand\lsup[1]{\prescript{#1}{}}

\makeatletter
\newcommand\topcite[1]{\gdef\t@pic{#1}\cite{#1}}
\newcommand\loccit{\expandafter\cite\expandafter{\t@pic}}
\makeatother


Suppose the depth of the representation $\pi$ is fixed. Then there are finitely many points in the alcove that can arise in J.-K. Yu's construction. This list will become part of fixed choices. In this section we describe this finite (and field-independent) set of points. 
\todo{Julia: I am not sure that defining the motivic building is the way to go; I do not really like the whole copy of 
$\R$ for every root sitting there; I wonder if we can simply describe a finite set of points in the building (baricentres of ...) such that any $x$ that can arise in J-K construction of a representation of $G$ of a given depth occurs in that set (just that: we do not need to make sure that every element of the set does occur). I think that's all we need from this section, we might not actually need the resulting paregorics, even, since we are not redoing the construction, just describing the parameters that go into it. In fact, I think here there's no need to do it motivically; we can use a $p$-adic field, just make a note that the construction does not depend on the field since the alcove, etc. does not. }


Let \(\bS\) be a maximal split torus in \(\bG\),
%There is a torus \(\bS^\sharp\) in \(C_\bG(\bS)\) such that
%\(\bS \times_\field \unfield\) is a maximal \(\unfield\)-split
%torus in \(\bG \times_\field \unfield\).
%Then \(\AA(\bS, \field)\) embeds naturally in
%\(\AA(\bS^\sharp, \unfield)\).
and \(C\) an alcove in \(\AA(S)\).
%and \(C^\sharp\) an alcove in \(\AA(\bS^\sharp, \unfield)\)
%containing \(C\).
Write \(\affW(G, S; C)\) for the (finite)
set of elements \(w \in \affW(G, S)\) such that
\(w\dota C\) is adjacent to \(C\),
and put
\[
\lsub{S, C}\motB(G)
= \R^{\Root(G, S)} \times
	2^{\affW(G, S; C)}.
\]
This should be regarded as a very loose analogue of the
\emph{reduced}, not the full enlarged, Bruhat--Tits building
of \(G\)
(or, better, of its quotient by the action of \(G\)).

If \(\xi = (t, W') \in \lsub{S, C}\motB(G)\) and \(r \in \R\),
then we define
\(\fg_{\xi, r} \ldef \lsub{S, C}\fg_{\xi, r}\)
and (if \(r \ge 0\))
\(G_{\xi, r} \ldef \lsub{S, C}G_{\xi, r}\)
as follows.
Let \(\AffRoot_{\xi, r}\) be the set of affine roots \(\psi\)
for which
\(\psi \ge \psi_\alpha + (r - t_\alpha)\),
where \(\alpha = \dot\psi\) is the gradient of \(\psi\)
and \(\psi_\alpha\) is the smallest affine root with
that gradient that is positive on \(C\)
\topcite{moy-prasad:k-types}*{\S2.5};
and put
\[
\fg_{\xi, r}
= \Cent_\fg(S)_r \oplus
	\sum_{\psi \in \Psi_{\xi, r}} \mathfrak u_\psi
\quad\text{and}\quad
G_{\xi, r}
= \langle
	\Cent_G(S)_r, U_\psi\st \psi \in \Psi(\xi, r)
\rangle
\]
\loccit*{\S2.4}.
	\Lxxx{Be careful with the MP references (in this
case, to the definitions of `affine root' and of the root
subgroup associated to an affine root); they are defined
over \(\unfield\).  I \emph{think} that the statement is
true anyway; or we could work over \(\unfield\) and then
perform Galois descent
(which would require changing the definition of
\(\motB(G)\)).}
Also, put
\[
\stab_G(\xi) \ldef \lsub{S, C}{\stab_G(\xi)}
	= \langle G_{\xi, 0}, W'\rangle.
\]
Note that \(G_{\xi, 0}\) contains \(\Cent_G(S)_0\), so that
there is no concern about which representative in
\(\Norm_G(S)\) we choose for an element of \(\affW(G, S)\).
The notation is unfortunate, since there is no obvious
action of \(G\) on \(\lsub{S, C}\motB(G)\).

Now we define a map
\(\BB(G) \to \lsub{S, C}\motB(G)\),
as follows.
For any \(x \in \BB(G)\), there is a unique point \(x'\) in the
intersection with \(G\dota x\) of the closure of \(C\)
	\cite{bruhat-tits:reductive-groups-1}*{Corollaire 2.1.6}.
Say that \(g \in G\) is such that \(g\dota x = x'\).
We associate to \(x\) the pair
\(\xi \ldef \xi_{S, C}(x) = (t, W')\), where
	\begin{itemize}
	\item for each \(\alpha \in \Root(\bG, \bS)\),
\(t_\alpha\) is the value at \(x\) of the smallest affine
root with gradient \(\alpha\) that is positive on \(C\),
and	\item \(W'\) is the set of all those elements of
\(\affW(G, S)\) that fix \(x'\) (hence lie in \(\affW(G, S; C)\)).
	\end{itemize}
Note that
\(\fg_{\xi, r} = \fg_{x', r} = \Ad(g\inv)\fg_{x, r}\)
and (if \(r \ge 0\))
\(G_{\xi, r} = G_{x', r} = \Int(g\inv)G_{x, r}\)
for all \(r \in \R\);
	\Lxxx{Cite MP (or a non-quasi-split generalisation)
for parahoric subgroups.}
and \(\stab_G(\xi) = \stab_G(x') = \Int(g\inv)\stab_G(x)\)
	\cite{bruhat-tits:reductive-groups-2}*
		{Proposition 4.6.28(ii) and \S5.1.31}.

Suppose that \(\bS'\) is another maximal split torus in
\(\bG\), and \(C'\) is a chamber in \(\AA(S')\).
There exist elements \(g \in G\) such that
\[
\Int(g)\bS = \bS'
\quad\text{and}\quad
g\dota C = C';
\]
	\cite{bruhat-tits:reductive-groups-1}*
		{Corollaires 2.3.7(ii) and 2.3.3}
and, for any such element, conjugation induces a root-system
isomorphism
\(\Root(\bG, \bS) \to \Root(\bG, \bS')\)
and a group isomorphism
\(\affW(G, S) \to \affW(G, S')\)
that carries \(\affW(G, S; C)\) to \(\affW(G, S'; C')\),
hence a bijection
\(\lsub{S, C}\motB(G) \to \lsub{S', C'}\motB(G)\).
If the bijection carries \(\xi \in \lsub{S, C}\motB(G)\) to
\(\xi' \in \lsub{S', C'}\motB(G)\) and \(r \in \R\), then
\[
%\begin{align*}
\lsub{S', C'}\fg_{\xi', r}
%& {}
= \Ad(g)\lsub{S, C}\fg_{\xi, r}
% \\
%\lsub{S', C'}G_{\xi', r}
%& {}= \Int(g)\lsub{S, C}G_{\xi, r} \\
%\intertext{(if \(r \ge 0\)) and}
%\lsub{S', C'}\stab_G(\xi')
%& {}= \Int(g)\lsub{S, C}\stab_G(\xi);
%\end{align*}
\]
(and so on);
and, for \(x \in \BB(G)\), the
bijection carries \(\xi_{S, C}(x)\) to \(\xi_{S', C'}(x)\).

Thus, there is no ambiguity in writing just \(\motB(G)\);
and, for any \((\xi, r) \in \motB(G) \times \R\),
in writing \(\fg_{\xi, r}\) (and so on), as long as we care
about the mentioned group only up to (simultaneous)
\(G\)-conjugacy.


\subsection{Deligne-Lusztig representations} 
For now, consider the case when $\rho_0$ is a Deligne-Luszting representation. 
Then it is determined by an elliptic torus $\fT$ in $\fG$, which is determined by an element $w$ in the Weyl group of $\fG$, and a character $\chi$ of $\fT$.  
The element $w$ becomes  part of the fixed choices.
  
The multiplicative characters $\phi_i$, as well as $\chi$, will play no role, so we will group 
representations into families with the same $(\vec G, x, w)$. 

Thus, fixed choices determine: the root system of $G$, root systems of $G_i$, 
the point $x$ that determines the parahoric $K_0$, and an element $w$ that determines 
$\fT$.  
Every fixed choice gives a family of representations of each of the groups $G_z$, with $z\in Z$ -- the cocycle space determined by this fixed data.
For a given fixed choice, we have the cocycle spaces $Z$ (determining $G$), and $Z^i$ (determining $G^i$), and from this we can reconstruct the representations. 
Let $\cF_d$ be the set of fixed choices for fixed $d$.
The elements of $\cF_d$ are tuples 
$\left((X^\ast, X_\ast, \Phi, \Phi^\vee), \Sigma, (\Phi^i, {\Phi^\vee}^i)_{i=0}^d, \Sigma^i, x, w\right)$, where 
\begin{enumerate}
\item  $(X^\ast, X_\ast, \Phi, \Phi^\vee)$  determines the absolute root datum of $G$, 
\item $\Sigma$ is an enumerated Galois group (with a choice of inertia subgroup and a 
generator of its cyclic quotient that gives the Galois group of the maximal unramified 
subextension) of the Galois extension that splits $G$,
\item  $(\Phi^i, {\Phi^\vee}^i, \Sigma^i)$ does the same for $G^i$. 
\item  The point $x$  determines the parahoric $K^0$.   
\item $w$ determines  $\fT$. 
\end{enumerate}

We want to prove: 
\begin{theorem} (Deligne-Lusztig case) Fix an element of $\cF_d$. 
Then there is an associated cocycle space $Z\times Z^1\times \dots Z^d$, and a family of representations $\pi_{z_i}$ of $G_z$. 
For $F\in \Loc_m$, every connected reductive group $G$ over $F$ appears as a member  of such  family for some fixed choice; 
every supercuspidal representation of $G$ that has a Deligne-Luzstig depth-zero component 
appears in the family of representations of $G_z$, and 
the coefficients of Harish-Chandra's local character expansion are motivic functions on 
$Z\times Z^1\times \dots \times Z^d\times \NF_G^k$, where $k$ is the number of nilpotent orbits in $\fg_z$. 
\end{theorem}

General case: should include the decomposition of a general character near the identity into Deligne-Lusztig ones, and the parameters that account for this -- will probably be part of fixed choices. 

\subsection{field-independent choices for the depth-zero data}




\section{Asymptotic character formulae}

\subsection{Depth-zero characters}
\subsubsection{Deligne--Lusztig case} Let us fix an element of the set of fixed choices $\cF$. 
That determines the cocycle space $Z$ whose elements give rise to the groups $G_z$. This fixed 
choice also contains a point in the building of $G$ that gives rise to a maximal parahoric  $K^0$
 with reductive quotient $\fG$, and an elliptic torus $\fT$ in $\fG$. 
Given a cuspidal character $\chi$ of $\fT$, there is an associated irreducible Deligne--Lusztig representation $R_{\fT, \chi}^\fG$. Let $\Theta_{\fT, \chi, x}$ be the character of the associated depth-zero representation of $G$.
In the proposition below, everything depends on our fixed choice, but we drop it from the notation in order to lighten it. 
\begin{prop} There exists $m$, and 
%positve integers $m$ and $a$, and 
a motivic function $C$ on $Z\times \cN_{G}$, such that for every $F\in \Loc_m$, 
for all $z\in Z(F)$, for every $\chi$ -- a multiplicative character of the residue field of $F$, 
for every $g\in \fg_{0^+}$, we have, for every $(N, \ff)=(N_i, \ff_i)_{i=1}^k\in NF_G^k$,  
$$\theta_{\fT, \chi, x}(\mexp(Y)) = \sum_{i=1}^k C_F(N_i)\widehat \cO_{N_i}(Y).$$
(Recall that the representation of $G_z$ whose character we are considering is determined by the data that is part of the  fixed choice that we have specified at the beginning of this subsection).  
\end{prop}

\subsubsection{The general case}

\subsection{mock exponential maps}
Need to have a definable set of parameters that give possible choices of the mock exponential map that we care about (a change of $\mexp$ can affect the coefficients a little). 

\subsection{asymptotic character expansion}

Yu associates to a datum \((\vec G, \vec\phi, \pi_0)\)
satisfying certain conditions
\topcite{yu:supercuspidal}*{p.~590, \textbf{D1}--\textbf{D5}}
a supercuspidal representation \(\pi\) \loccit*{\S4}.
\textit{Via} the Moy--Prasad theory
\cite{moy-prasad:jacquet}*{Theorem 6.8}
(and \cite{yu:supercuspidal}*{Lemma 3.3}), there is
associated to the depth-0 representation \(\pi_0\) a vertex
\(x_0\) of \(\BB(G^0)\).
Kim and Murnaghan \topcite{jkim-murnaghan:gamma-asymptotic}
associate to the strongly good, positive \(G\)-datum
\((\vec G, x_0, \vec\phi)\) \loccit*{Definition 4.1.1, 3}
an element \(\Gamma\) in \(\fg\) \loccit*{Definition 4.1.3},
and prove that, if \(\cO^G(\Gamma)\) is the set of adjoint
orbits of \(G\) on \(\fg\) whose closures contain \(\Gamma\)
\loccit*{\S2.1.4}, then there is an indexed family
\((c_\cO(\pi))_{\cO \in \cO^G(\Gamma)}\) of complex numbers
such that
\[
\Nchar_\pi(\mexp(Y))
= \sum_{\cO \in \cO^G(\Gamma)}
	c_\cO(\pi)\NFTorbit^G_\cO(Y)
\]
for \(Y \in \fg_{r/2}\), where \(r\) is the minimal depth of
\(\pi\)
\loccit*{Theorem 4.4.1}.
(Technically, their definition of a good, positive
\(G\)-datum includes an additional piece of information
\(\vec r\), but this is just the vector of depths of the
characters \(\vec\phi\).)
The following theorem appears in \cite{spice:coefficients}.

\begin{theorem}
We have that
\[
\Nchar_{\pi_0}(\mexp(Y))
= \sum_{\cO_0 \in \cO^{G^0}(0)}
	c_{\Ad(G)(\Gamma + \cO^0)}(\pi)\NFTorbit^{G^0}_{\cO^0}(Y)
\]
for \(Y \in \fg^0_{0+}\).
	\Lxxx{Check whether this should be at depth \(r+\)
(to avoid worries about twisting by characters).  I think
that it is taken care of by working with \(\cO^{G^0}(0)\)
instead of \(\cO^{G^0}(\Gamma)\).}
\end{theorem}

That is, the coefficients in the local character expansion
of \(\pi_0\) are the same as the corresponding coefficients
in the asymptotic character expansion of \(\pi\).
Since
\(\Cent_G(\Gamma) = G^0\) \citeme,
the `fattening map'
\(\cO^{G^0}(0) \to \cO^G(\Gamma)\)
that sends \(\cO^0\) to \(\Ad(G)(\Gamma + \cO^0)\) is
surjective \cite{hc:queens}*{Lemma 2.8},
so that the asymptotic character expansion of
\(\pi\) contains precisely the same information as the local
character expansion of \(\pi^0\).
\Lxxx{Here, \(\Nchar\) stands for the `normalised' character
(by the square root of the Weyl discriminant of its
argument), and \(\NFTorbit\) stands for the `normalised' FT
of an orbital integral (by the square roots of the Weyl
discriminants of its argument \emph{and} the `orbiting'
element).
Unfortunately, the current notation for \(\Nchar\) conflicts
with the notation for a root system.
Notice that the normalisation for the FT of a
\emph{nilpotent} orbital integral involves only the
discriminant of its argument.

In \cite{spice:coefficients}, the `peeling' process from
\(\pi\) to \(\pi'\) keeps the multiplicative character
\(\phi'\) on \(\pi'\), so that we would expect our \(\pi_0\)
to be twisted by \(\prod_{i = 0}^{d - 1} \phi_i\) and our
\(\cO^{G^0}(0)\) to be replaced by \(\cO^{G^0}(\Gamma)\);
but these corrections (I think) cancel each other out,
leaving us with the tidy statement above.}
 
\begin{theorem} The coefficients $c_{\cO}$ of this expansion are motivic functions of the parameters defining $G$, $\pi$ (as above) and $\mexp$. 
\end{theorem}

\begin{proof}
\end{proof}

\begin{cor}
Coefficients of Harish-Chandra local character expansion are motivic functions of the same thing.
\end{cor}

\section{Done!}

\begin{bibdiv}
\begin{biblist}
\bibselect{spice-references}
\end{biblist}
\end{bibdiv}
\end{document}
